A physics teacher: uncertainty

This is the second part of a series of post on finding uncertainty in derived quantities. You can find the index here and the part on addition and subtraction here.

Now very often when you performed an experiment you meet quantities that need to be processed to obtain a derived quantity such as g, the acceleration due to gravity, or any other derived quantities.

You have already seen the first part on addition and subtraction and now you will see how to do it for multiplication and division.

If in an experiment is performed and the following quantities are measured with their uncertainties.

A = 10.2+-0.2 cm

B = 5.4 +-0.4 cm

Now if you need to process these quantities to find AB and A/B with their uncertainties how would you do it?

Uncertainty in multiplication

How would you determine the value of AB and its uncertainty?

You will have to calculate the value of AB first.

AB = 10.2 *5.4 = 55.o8 =55 (2 sf )

To determine the uncertainty in AB you will have to use the equation below.

Rearranging the equation will give you

Hence AB = 55+-5 cm²

Remember the uncertainty in A and B are to 1 sf hence the uncertainty in AB must be given to 1 sf.

Uncertainty in division

You are now going to determine the value of A/B and its uncertainty.

You will have to determine the value of A/B first.

A/B = 10.2 /5.4 =1.888 = 1.9

To determine the uncertainty of A/B you will determine the equation below

Rearranging the equation will give you

Remember the uncertainty in A/B is given to 1 sf since the uncertainties in A and B are given to 1 sf.

Hence A/B = 1.9 +-0.2

With these two formula you can thus determine the uncertainty in any derived quantities that involves multiplication and division.

Now you can move to the next part where the method to determine uncertainty of derived quantities where powers, square root, etc are involved.

As we have seen in an earlier post, a physical quantity is a property of an object that can be measured with a measuring instrument. Hence when you use a measuring instrument you would obtain a reading. The reading is simply a numerical value that you can read off a measuring instrument such as the volume off a measuring cylinder or the time off a stopwatch.

We must also introduce what is called the true value. The true value is the reading that you would obtain if the measurement is done in ideal conditions.

In order to obtain the the true value the following conditions must be present:

You must have the skills to use the instrument and know the steps that must be followed to obtain the reading.
You must be using instruments that are properly calibrated and are not damaged.
All the conditions that are required to obtain the true value must be present. For example 1/3 of the stem of the thermometer must be immersed in the liquid, the pressure must be one bar, the liquid used in the measuring cylinder must have a temperature of 20⁰C.
If a calculation is needed to determine the magnitude of a physical quantity, then the correct equation and the right constant must be used.

Now an error is made when the reading that you obtain is not equal to the true value. And the magnitude of the error is the difference between the reading obtained and the true value.

Then we can thus introduce the term accuracy and precision.

A measurement is accurate if it is close to the true value. And if there are two values then the one that is closest to the true value is the most accurate.

The precision of an instrument is the smallest value that can be measured using the instrument. Hence if a length is measured using a metre rule then it is precise to 1 mm or it has a precision of 1 mm. However a length that is measure using a vernier caliper has a precision of 0.1 mm. Hence the length that is measure using a vernier caliper is most precise.

Thursday, November 26, 2009

Measurement, errors and uncertainties

Calculation

Calculation in Physics

Performing addition and subtraction

Performing multiplication and division

Performing Calculation with lg and ln

Uncertainties and errors

Uncertainty and How to process it

How to determine uncertainty in a derived quantity when addition or subtraction is performed?

How to determine uncertainty in derived quantity when multiplication or division is performed.

How to determine uncertainty in a derived quantity when powers, root,etc are involved.

Saturday, November 21, 2009

How to determine uncertainty in a derived quantity when powers, root,etc are involved.

As we have seen in the first two part of this series it is easier to determine the uncertainty when addition and subtraction is involved but also to determine uncertainty when multiplication and division is involved.

Today we are going to see how to determine the uncertainty when a power is involved.

Very often a derived quantity is determined basic or othere derived quantities using powers, roots, etc.

For example

As we are going to see the method to determine the uncertainty in these three cases are similar and can be adapted to other derived quantities.

Example

Let us have a look at how to determine the uncertainty in volume.

A cube has length of L = 20.4+_0.2cm

Determine the uncertainty in volume.

V = L³

V = L*L*L

=249 = 200 (1 sf because of 0.2 )

V = L³

=20.4³

=8489

=8500 (2 sf because of 200)

Hence V = 8500+_200 cm³

After you have studied this example are you able to know how to find the uncertainty in any other derived quantities?

Then the fractional uncertainty is

What if area

Then can you deduce the fractional uncertainty?

Yes it is

Can you deduce what would be the fractional uncertainty if?

Yes it is )

Lastly you deduce the fractional uncertainty if ?

Where K is a dimensionless constant with no uncertainty.

Yes it is also )

When there is a dimensionless constant with no uncertainty it does not enter in the equation to calculate the fractional uncertainty.

Question

The area of a circle is given by the equation

If the radius r =10.1+_0.5 cm

then calculate the fractional uncertainty and the uncertainty in A.

Please not that is a dimentionless constant with no uncertainty and hence will play no part in thecalculations.

Monday, October 5, 2009

How to determine uncertainty in derived quantity when multiplication or division is performed

This is the second part of a series of post on finding uncertainty in derived quantities. You can find the index here and the part on addition and subtraction here.

Now very often when you performed an experiment you meet quantities that need to be processed to obtain a derived quantity such as g, the acceleration due to gravity, or any other derived quantities.

You have already seen the first part on addition and subtraction and now you will see how to do it for multiplication and division.

If in an experiment is performed and the following quantities are measured with their uncertainties.

A = 10.2+-0.2 cm

B = 5.4 +-0.4 cm

Now if you need to process these quantities to find AB and A/B with their uncertainties how would you do it?

Uncertainty in multiplication

How would you determine the value of AB and its uncertainty?

You will have to calculate the value of AB first.

AB = 10.2 *5.4 = 55.o8 =55 (2 sf )

To determine the uncertainty in AB you will have to use the equation below.

Rearranging the equation will give you

Hence AB = 55+-5 cm²

Remember the uncertainty in A and B are to 1 sf hence the uncertainty in AB must be given to 1 sf.

Uncertainty in division

You are now going to determine the value of A/B and its uncertainty.

You will have to determine the value of A/B first.

A/B = 10.2 /5.4 =1.888 = 1.9

To determine the uncertainty of A/B you will determine the equation below

Rearranging the equation will give you

Remember the uncertainty in A/B is given to 1 sf since the uncertainties in A and B are given to 1 sf.

Hence A/B = 1.9 +-0.2

With these two formula you can thus determine the uncertainty in any derived quantities that involves multiplication and division.

Now you can move to the next part where the method to determine uncertainty of derived quantities where powers, square root, etc are involved.

How to determine uncertainty in a derived quantity when addition or subtraction is performed?

As we have seen in the previous post it is important to know how to determine the uncertainty for derived quantities.

Let us see how to determine how to determine the uncertainty of a derived quantity when an addition or a subtraction is performed.

In an experiment two quantities are measured as shown.

A = 56.4 +_0.1 cm

B = 12.2+_0.1 cm

If a derived quantity C is given by the equation below

C = A + B

Then the uncertainty in C C = A +B

C = 0.1 + 0.1

= o.2 cm

C = 56.4 + 12.2 = 68.6 cm

C = 68.6 +-0.2 cm

If you want to calculate C and the equation relating C to A and B is

C = A – B

Then we can use the following equation to calculate C

C = A +B

Hence C = 0.1 + 0.1 = 0.2 cm

If C = 56.4 – 12.2 = 44.2 cm

then C = 44.2 +-0.2 cm

This is then end of this part. Please follow the link below to go to the part on determining the

Saturday, October 3, 2009

Uncertainty and how to process it!

In physics all quantity that is measured come with its uncertainty. This is because the instrument, the experiment or the person performing the measurement is subjected to some limitations.

For example

Someone that is measuring a length with a metre rule cannot measure the length to a precision that is better than one mm. So a length that is measured with a metre rule will have an uncertainty of +-o.1 c m.
The limitation can either be due to the experiment itself, the apparatus, or the experimenter. Someone measuring the height of rebound will measure it with an uncertainty of +- 0.5 cm. Someone estimating the mass of an object can measure it with an uncertainty of +-0.5 kg. A voltmeter will give a reading with a percentage uncertainty of 1%.

[As you may have guess from 2 above not all measurements that are performed will have these uncertainty. These are for example only.]

As you can see all quantities that are measured will have uncertainties. And as a result any other quantities that are derived from the measured quantities will have their uncertainties.

We are going to see how to determine the uncertainty when the following operations are performed.